Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schroedinger maps on R^2

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schroedinger flow as special cases) for degree m equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal energy solutions converge to a harmonic map as t goes to infinity (asymptotic stability), extending previous work down to degree m = 3. Due to slow spatial decay of the harmonic map components, a new approach is needed for m=3, involving (among other tools) a "normal form" for the parameter dynamics, and the 2D radial double-endpoint Strichartz estimate for Schroedinger operators with sufficiently repulsive potentials (which may be of some independent interest). When m=2 this asymptotic stability may fail: in the case of heat-flow with a further symmetry restriction, we show that more exotic asymptotics are possible, including infinite-time concentration (blow-up), and even "eternal oscillation".Comment: 34 page

Non-universal scaling and dynamical feedback in generalized models of financial markets

We study self-organized models for information transmission and herd behavior in financial markets. Existing models are generalized to take into account the effect of size-dependent fragmentation and coagulation probabilities of groups of agents and to include a demand process. Non-universal scaling with a tunable exponent for the group size distribution is found in the resulting system. We also show that the fragmentation and coagulation probabilities of groups of agents have a strong influence on the average investment rate of the system

Understanding complex dynamics by means of an associated Riemann surface

We provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated Riemann surface. Thanks to this geometric description an explicit formula for the period of the orbits can be derived, which is shown to depend on the initial data and the continued fraction expansion of a simple ratio of the coupling constants of the problem. For rational values of this ratio and generic values of the initial data, all orbits are periodic and the system is isochronous. For irrational values of the ratio, there exist periodic and quasi-periodic orbits for different initial data. Moreover, the dependence of the period on the initial data shows a rich behavior and initial data can always be found such the period is arbitrarily high.Comment: 25 pages, 14 figures, typed in AMS-LaTe

Orbit spaces of free involutions on the product of two projective spaces

Let $X$ be a finitistic space having the mod 2 cohomology algebra of the product of two projective spaces. We study free involutions on $X$ and determine the possible mod 2 cohomology algebra of orbit space of any free involution, using the Leray spectral sequence associated to the Borel fibration $X \hookrightarrow X_{\mathbb{Z}_2} \longrightarrow B_{\mathbb{Z}_2}$. We also give an application of our result to show that if $X$ has the mod 2 cohomology algebra of the product of two real projective spaces (respectively complex projective spaces), then there does not exist any $\mathbb{Z}_2$-equivariant map from $\mathbb{S}^k \to X$ for $k \geq 2$ (respectively $k \geq 3$), where $\mathbb{S}^k$ is equipped with the antipodal involution.Comment: 14 pages, to appear in Results in Mathematic

Determining the structure of the bacterial voltage-gated sodium channel NaChBac embedded in liposomes by cryo electron tomography and subtomogram averaging

Voltage-gated sodium channels shape action potentials that propagate signals along cells. When the membrane potential reaches a certain threshold, the channels open and allow sodium ions to flow through the membrane depolarizing it, followed by the deactivation of the channels. Opening and closing of the channels is important for cellular signalling and regulates various physiological processes in muscles, heart and brain. Mechanistic insights into the voltage-gated channels are difficult to achieve as the proteins are typically extracted from membranes for structural analysis which results in the loss of the transmembrane potential that regulates their activity. Here, we report the structural analysis of a bacterial voltage-gated sodium channel, NaChBac, reconstituted in liposomes under an electrochemical gradient by cryo electron tomography and subtomogram averaging. We show that the small channel, most of the residues of which are embedded in the membrane, can be localized using a genetically fused GFP. GFP can aid the initial alignment to an average resulting in a correct structure, but does not help for the final refinement. At a moderate resolution of ˜16 Å the structure of NaChBac in an unrestricted membrane bilayer is 10% wider than the structure of the purified protein previously solved in nanodiscs, suggesting the potential movement of the peripheral voltage-sensing domains. Our study explores the limits of structural analysis of membrane proteins in membranes

Non-commutative oscillator with Kepler-type dynamical symmetry

A 3-dimensional non-commutative oscillator with no mass term but with a certain momentum-dependent potential admits a conserved Runge-Lenz vector, derived from the dual description in momentum space. The latter corresponds to a Dirac monopole with a fine-tuned inverse-square plus Newtonian potential, introduced by McIntosh, Cisneros, and by Zwanziger some time ago. The trajectories are (arcs of) ellipses, which, in the commutative limit, reduce to the circular hodographs of the Kepler problem. The dynamical symmetry allows for an algebraic determination of the bound-state spectrum and actually extends to the conformal algebra o(4,2).Comment: 10 pages, 3 figures. Published versio

Hexagonal dielectric resonators and microcrystal lasers

We study long-lived resonances (lowest-loss modes) in hexagonally shaped dielectric resonators in order to gain insight into the physics of a class of microcrystal lasers. Numerical results on resonance positions and lifetimes, near-field intensity patterns, far-field emission patterns, and effects of rounding of corners are presented. Most features are explained by a semiclassical approximation based on pseudointegrable ray dynamics and boundary waves. The semiclassical model is also relevant for other microlasers of polygonal geometry.Comment: 12 pages, 17 figures (3 with reduced quality

Light-Front Quantisation as an Initial-Boundary Value Problem

In the light front quantisation scheme initial conditions are usually provided on a single lightlike hyperplane. This, however, is insufficient to yield a unique solution of the field equations. We investigate under which additional conditions the problem of solving the field equations becomes well posed. The consequences for quantisation are studied within a Hamiltonian formulation by using the method of Faddeev and Jackiw for dealing with first-order Lagrangians. For the prototype field theory of massive scalar fields in 1+1 dimensions, we find that initial conditions for fixed light cone time {\sl and} boundary conditions in the spatial variable are sufficient to yield a consistent commutator algebra. Data on a second lightlike hyperplane are not necessary. Hamiltonian and Euler-Lagrange equations of motion become equivalent; the description of the dynamics remains canonical and simple. In this way we justify the approach of discretised light cone quantisation.Comment: 26 pages (including figure), tex, figure in latex, TPR 93-

Connected Green function approach to ground state symmetry breaking in Φ1+14\Phi^4_{1+1}-theory

Using the cluster expansions for n-point Green functions we derive a closed set of dynamical equations of motion for connected equal-time Green functions by neglecting all connected functions higher than $4^{th}$ order for the $\lambda \Phi^4$-theory in $1+1$ dimensions. We apply the equations to the investigation of spontaneous ground state symmetry breaking, i.e. to the evaluation of the effective potential at temperature $T=0$. Within our momentum space discretization we obtain a second order phase transition (in agreement with the Simon-Griffith theorem) and a critical coupling of $\lambda_{crit}/4m^2=2.446$ as compared to a first order phase transition and $\lambda_{crit}/4m^2=2.568$ from the Gaussian effective potential approach.Comment: 25 Revtex pages, 5 figures available via fpt from the directory ugi-94-11 of [email protected] as one postscript file (there was a bug in our calculations, all numerical results and figures have changed significantly), ugi-94-1

Tunneling and propagation of vacuum bubbles on dynamical backgrounds

In the context of bubble universes produced by a first-order phase transition with large nucleation rates compared to the inverse dynamical time scale of the parent bubble, we extend the usual analysis to non-vacuum backgrounds. In particular, we provide semi-analytic and numerical results for the modified nucleation rate in FLRW backgrounds, as well as a parameter study of bubble walls propagating into inhomogeneous (LTB) or FLRW spacetimes, both in the thin-wall approximation. We show that in our model, matter in the background often prevents bubbles from successful expansion and forces them to collapse. For cases where they do expand, we give arguments why the effects on the interior spacetime are small for a wide range of reasonable parameters and discuss the limitations of the employed approximations.Comment: 29 pages, 8 figures, typos corrected, matches published versio